SAT Math: Find the Radius of Circle Defined by x² + y² - 14x + 10y = 7

Exact Question Text

The equation x^2 + y^2 - 14x + 10y = 7 defines a circle in the xy-plane. What is the radius of the circle?

1. Step-by-Step Formal Solution

1. Group: (x^2 - 14x + 49) + (y^2 + 10y + 25) = 7 + 49 + 25 = 81.
2. (x - 7)^2 + (y + 5)^2 = 81 => r = 9.

2. Fast Desmos / Calculator Shortcut

Type x^2 + y^2 - 14x + 10y = 7 in Desmos. Measure diameter (18) / 2 = 9.

3. Distractor Trap Analysis

• Trap A (Sign Flip): Forgetting to reverse inequality or dropping negative sign.
• Trap B (Partial Evaluation): Solving for intermediate variable and stopping early.
• Trap C (Extreme Assumption): Over-inferring beyond textual evidence.
• Option Correct: Accurately satisfies all algebraic and logical constraints.